Full-Speed Testing of A/D Converters
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1 820 EEE JOURNAL OF SOLD-STATE CRCUTS, VOL. SC-19, NO. 6, DECEMBER 1984 Full-Speed Tesing of A/D Converers JOEY DOERNBERG, STUDENT MEMBER, EEE, HAE-SEUNG LEE, STUDENT MEMBER, EEE, AND DAVD A. HODGES, FELLOW, EEE Absrac High-precision analog-o-digial converers (ADC S) are sough for digial audio and insrumenaion and high-speed converers for video applicaions. mproved mehods of converer esing a full speed are needed. This paper describes improved compuer-aided ADC characerizaion mehods based on he code densiy es and specraf analysis using he fas Fourier ransform (FFT). The code densiy es produces a hisogram of he digial oupu codes of an ADC sampling a known inpu. The code densiy can be inerpreed o compue he differenial and inegral nonlineariies, gain error, offse error, and inernal noise. Conversion-rae and frequency-dependen behavior can also be measured.. NTRODUCTON T HS paper describes improved compuer-based mehods of esing high-precision and high-speed analogo-digial converers (ADC S) a full speed wih full-range dynamic inpus. A known periodic inpu is convered by an ADC under es a sampling imes ha are asynchronous relaive o he inpu signal. The relaive number of occurrences of he disinc digial oupu codes is ermed he code densiy. These daa are viewed in he form of a normalized hisogram showing he frequency of occurrence of each code from zero o full scale. The code densiy daa are used o compue all bi ransiion levels. Lineariy, gain, and offse errors are readily calculaed from a knowledge of he ransiion levels. This provides a complee characerizaion of he ADC in he ampliude domain. The precision of his measuremen may be exended wihou limi by aking addiional daa. Oupu samples from an ADC also may be processed wih a fas Fourier ransform (FFT) algorihm o define he lineariy and noise properies of he ADC in he frequency domain. This is analogous o he use of analog specrum analysis o es digial-o-analog (D/A) converers. For an ideal ADC, he code densiy is independen of conversion rae and inpu frequency. The characerisics of pracical ADC S (wih heir associaed sample/hold circuis) can be exhausively esed by varying boh he sampling frequency and inpu frequency. Overall frequency Manuscrip received; revised July 23, This work was suppored by he Semiconducor Research Corporaion under Gran SRC and he Naionaf Science Foundaion under Gran ECS J. Doernberg and D. A. Hodges are wih he Deparmen of Elecrical Engineering and Compuer Sciences and he Elecronics Research Laboraory, Universiy of Cafifomla, Berkeley, CA H.-S. Lee was wih he Deparmen of Elecrical Engineering and Compuer Sciences and he Elecronics Reseach Laboraory, Uruversiy of California, Berkeley, CA He is now wih he Deparmen of Elecrical Engineering and Compuer Science, Massachuses nsiue of Technology, Cambridge, MA response can be evaluaed using he code densiy es for several inpu frequencies. n high-precision converers ( > 12 bis), noise is a major concern. The saisical naure of he code densiy es gives a more accurae characerizaion of converer noise compared o convenional ess in which each oupu code is aained only once. Noise ampliude can be compued in rms, peak, or specral (from FFT) form. Tradiional ess use a digial volmeer (DVM) o aain high measuremen accuracy, bu he ess are done wih a saic or slowly varying inpu signal. A dynamic inpu can be creaed using a digial-o-analog converer (DAC), bu i is difficul o separae he errors of he DAC and ADC. Furhermore, resoluion is limied; esing a 16 bi ADC wih an 18 bi DAC (if i exiss) only yields 1/4 bi precision in he ADC es.. CODE DENSTY TEST THEORY The hisogram or oupu code densiy is he number of imes every individual code has occurred. The firs observaion is ha an oupu code densiy or hisogram bin equal o O is a missing code. A shif in he densiy is an offse error. A change in slope of he ADC ransfer curve causes a gain error ha may be found by comparison o exernal ampliude measuremens. For an ideal ADC wih a full scale ramp inpu and random sampling, an equal number of codes is expeced in each bin. Differenial nonlineariy is he deviaion from one leas significan (LSB) of he range of inpu volages ha give he same oupu code. The number of couns in he ih bin H(i) divided by he oal number of samples N, is he widh of he bin as a fracion of full scale. The raio of bin widh o he ideal bin widh P(i) is he differenial lineariy and should be uniy. Subracing one LSB gives he differenial nonlineariy in LSB S [1]: DN(i) = M(i)/N P(i) _l. (2.1) negral nonlineariy is he deviaion of he ransfer curve from idealiy. By compiling a cumulaive hisogram, he cumulaive bin widhs are he ransiion levels. Once he ransiions are known, he ADC is characerized. Overall noise is measured by grounding he ADC inpu and accumulaing a hisogram. Only he bin for zero inpu should have couns in i. Any oher couns are caused by noise in he ADC /84/ $ EEE
2 DOERNBERG e al.: FULL-SPEED TESTNG OF A/D CONVERTERS 821 By increasing he ADC conversion rae and comparing This is he probabiliy ha a code will be in bin [i] for an he desired properies such as lineariy, he maximum inpu sine wave of ampliude A. conversion rae for a desired accuracy can be deermined. f he inpu has a dc offse, i is of he form xc,+ Similarly, varying he inpu es frequency is a frequency A sin d wih densiy response measuremen, a rue dynamic es of he ADC. f an exernal sample-and-hold is used wih he ADC, i is P(v)= also being esed as par of he whole sysem. R + A (v vo) ) A. Choice of npu Waveform A firs glance, he choice for an inpu would be a ramp or riangle wave. An equal number of samples per bin is expeced, excep for he firs and las bins which would accumulae all couns for inpus ouside he converer s range. The fundamenal drawback o his is he disorion or nonlineariy in he ramp. For a differenial nonlineariy es, a 1 percen change in he slope of he ramp would change he expeced number of, codes by 1 percen. Bu hese errors would quickly accumulae and make he inegral nonlineariy es unfeasible. Brief consideraion makes i clear ha he inpu source mus be known wih beer precision han he converer being esed. A random volage wih an equal likelihood of all volages over a range is desired. Noice ha his is no whie noise which is equal ampliudes a all frequencies. A possible way o generae such a signal is o generae a pseudorandom digial sequence and hen use an analog low-pass filer o generae he random volage [2]. The drawback o his mehod is ha he digial sequence mus no change ampliude and he filer mus be ideal so as no o inroduce disorion. We have used a sine wave signal source. is precisely known mahemaically, and commercial ulralow disorion oscillaors have oal harmonic disorion < 95 db. This can be confirmed by a specral analysis. is much harder o measure he lineariy of a ramp o a comparable level of accuracy. 1) Sine Wave Probabiliy Densiy: The probabiliy densiy p(v) for a funcion of he form A sin a is (V)=AA (2.2) negraing his densiy wih respec o volage gives he disribuion funcion P(V=, V~): The new disribuion is jus shifed by V. as expeced from he shifed hisogram. P(V~, V5) = ; sm 1( -P= l-sin-r=v (2.6) The discree disribuion becomes l._l P(i, A, VO)=; sm { [( sin l B. Frequency of npu Waveform [( 2i 2 1 2V~ K-f 2n )-1 A Zi zn 3 2v0 vref 2n )-) A (2!.7) The foundaion of he esis ha a sine wave is sampled randomly. Sampling a random by is sric definiion would be impossible. Wha mus be done is o assure ha he sine wave inpu is no sampled repeiively a he same level. By choosing he sample frequency o be nonharmonically relaed o he sine wave frequency, we are assured of his. Any jier in he sample iming or drif in he oscillaor frequency will jus end o randomize he sampling. The effec of sampling a a frequency harmonically relaed o he inpu would be n bins wih huge posiive differenial nonlineariy where n is he raio of sample o inpu frequency. This can be easily disinguished from differenial nonlineariy by varying eiher he sample or inpu frequency since differenial nonlineariy is independen of frequency. For a high-speed converer, he conversion rae lmay exceed he rae a which a compuer can assemble he hisogram. is permissible o use very second or nh sample and hrow he exras away. Since he samples are aken a random, i does no maer if he firs M samples are used or if M ou of N samples are chosen. P(V., V~) =; sm (2.3) C. Number of Samples Needed 1( -f+sin-fw To find he minimum number of samples needed for This is he probabiliy of a sample being in he range V= o esimaing he differenial nonlineariy, a 100(1 a) per- Vb. cen confidence inerval of he form (p 2./20, P Z,,,@) For an ADC, le Vb Va= 1 bi and conver he coninu- is se up. This says ha he measured differenial nol nlineariy lies in he range (p Z.,2U, p + Za,2u) wih 1OO(1 ous probabiliy disribuion o a discree disribuion: a) percen probabiliy y. a is chosen for he desired confi- 2i 2 1 V,.f P(i, A) =; sin-l dence level. Z.,2U is he precision o which he measured ( [( )-1 value differs from he rue value p. The derivaion of c~and sin- [(2i-~-3)~]) 24) cariedouinheappendix. he subsequen minimum number of samples needed is
3 822 EEE JOURNAL OF SOLD-STATE CRCUTS, VOL. SC-19, NO. 6, DECEMBER 1984 PARALLEL LS-ll VAX NTERFACE l TERMNAL 1 Fig. 1. Experimenal seup for esing an ADC wih he code densiy es or FFT es. L 4 p.,, PREAD ADC- REAO ADC NTO ( N+ --N N S No OF SAMPLES TAKEN { The minimum number of samples N, needed for~ bi precision and 100(1 a) percen confidence is given by (2.8) where 2.,, can be found in a able of he sandard normal disribuion funcion: NO 9H[ 1+ -H[ N = MAX? [USE, AS A PONTER TO BN NCREMENT CONTENTS OF BN H[!] (2.8) To know he differenial nonlineariy for an 8 bi converer o wihin 0.10 bi wih 99 percen confidence, samples are needed. n he 12 bi case, for 99 percen confidence and 0.10 bi precision, 4.2 million samples are needed.. HARDWARE FOR A/D CONVERTER TESTNG The experimenal seup shown in Fig. 1 consiss of he inpu source, ADC sysem under es, a parallel inerface o an LS-11 minicompuer, and a VAX The ADC sysem consiss of he ADC, a sample-and-hold if needed, volage references, and conrol circuiry. Parallel daa from he ADC are lached and buffered on he inerface board before being read by he LS-11 hrough a parallel 1/0 por. The LS-11 was used o accumulae he daa since i had a parallel inpu por and 64K of 16 bi memory, enough o es a 16 bi ADC, and was available wih he Speechlab [3] program for digial 1/0, as well as communicaion wih a VAX running UNX. Any compuer can be used o compile he hisogram provided i has enough n-bi memory for 2 bins and he 1/0 hisogram program. Depending on he ime needed o compue he ADC ransiions and he availabiliy of a high-level language, he characerizaion could be compued on he same machine. n our work, once a hisogram is compleed, i is wrien o a UNX file on a VAX where he nonlineariy compuaions are carried ou. Fig. 2. Flowchar for hisogram accumulaing program running on he LS-11. ially in memory so only 64K samples could be aken. This is barely enough for esing an 8 bi ADC. The main modificaion was o use he digial code as a poiner o a memory locaion used as a couner as shown in Fig. 2. ncremening ha couner each ime i is accessed forms he hisogram. A fuure improvemen will be o wrie he daa inpu and hisogram rouine in Assembly language raher han C o improve upon he 9 khz daa inpu rae by approximaely a facor of 2. Program JADE does he ADC analysis from he hisogram daa and is shown funcionally in Fig. 3. is wrien in C and runs on a UNX sysem. The program firs ges command line argumens o se opions such as daa ype and oupu lisings. The user hen eners he name of he binary file conaining he hisogram. Nex he offse volage is compued and a cumulaive hisogram is compiled. From his, he ransiion levels are compued, leading o he nonlineariy calculaions. Once an ADC is characerized, he differenial nonlineariies and inegral nonlineariies can be wrien o ASC or binary files and ploed on a graphics erminal. A saisics file conains informaion such as he inpu offse, LSB size, and he maximum and minimum nonlineariies, ec. V. SOFTWARE FOR A/D CONVERTER TESTNG The sofware is used in wo sages. Speechlab is used o ake he hisogram and JADE o ~ompue he ADC errors. Speechlab is a general-purpose program wrien in C for an LS-11 o do analog 1/0 via an ADC and DAC, as well as digial 1/0 hrough a DRV-11 parallel 1/0 board. A modified version of Speechlab is used o gaher daa o es ADC S. Originally inpu daa were sored sequen- A. Algorihms for A /D Converer Error Compuaion The offse volage is found from he shif of he hisogram abou he midpoin O V. f VO= O, he number of codes above zero, NP, equals he number of codes below zero, NH:.2-1 Nn= ~ [i] NP= ~ [i]. (4.1),=1 i=*n l+~
4 DOERNBERG e al.: FULL-SPEED TESTNG OF A/D CONVERTERS 823 $SN START i READ n, FLENAME READ H[i] 1 ~-+Vo i ~ H[j] -c4i11 1,0 rch])_v[i] cos(~ Ml#!l.l-,)_lN[,] V[l+ll-v[ll LSB +DN[i] OUTPUT NONLNEARTES 1 GET PROGRAM OPTONS ENTER NUMBER OF BTS AND HSTOGRAM FLENAME READ. HSTOGRAM FLE TO H[, ] COMPUTE OFFSET VOLTAGE, VO COMPLE CUMULATVE HSTOGRAM, CH[, ] COMPUTE TRANSTON VOLTAGES, 4 1 COMPUTE NTEGRAL NONLNEARTY, N[i] COMPUTE DFFERENTAL NONLNEARTY DN[, ] WRTE DFFERENTAL AND NTEGRAL NONLNEARTY FLES Fig. 3. Flowchar forcode densiy analysis program which compucs he ADC ransiions from he hisogram. The probabiliypp ha any randomly sampled volage is posiive is he probabiliy ha i is in he range (O, A + VO) and is found from (2.6) o be P,=+ Sin- (l) -sin- ~ (4.2) { [ 1} (4.3) And he probabiliy p. ha a negaive volage is sampled is Solving (4.3) and (4.4) for V& VO=A~sin(pP p.). (4.5) An esimae of V& fio can be obained by replacing he unknown populaion frequencies pp and p. by he observed sample frequencies NP/N1 and N./N,: io=a~sin Np Nn NP + N. (4.6) where NP and N. are he number of posiive and negaive samples, respecively. When he offse volage is small relaive o he sine ampliude, his can be approximaed n compuing he differenial nonlineariy, subsiuing (2.7) ino (2.1) for P(i) is unfeasible and incorrec. (4.7) is unfeasible since he ampliude of he sine wave A mus be known ih grea precision because he differenial nonlineariy calculaion is a very srong funcion of A. To see he accuracy and precision o which A mus be known, assume a perfec ADC. Now if A is hough o be equal o full scale, a cerain number of codes is expeced in bin [1] and bin [2 ]. Bu if A is jus A 1/2 LSB, approximaely 1/2 as many codes will be obained and he differenial nonlineariy will be 1/2 bi in hese wo bins. When oo few codes go ino hese wo bins, oher bins ge he exra codes, resuling in excess posiive differenial nonlineariy. Vr,f, he full-scale volage reference, is needed, bu being a dc quaniy, i can be measured wih a DVM o sufficien precision. The erm A, however, is he peak volage wih a dc offse, no an rms volage, and is measured less accuraely wih a DVM han a dc volage. Mos DVM S n[leasure ac quaniies a 60 Hz and do no have he bandwidh o measure A a a few kiloherz. The second consideraion is due o he nonlineariy of he sine wave. Twice as many codes are no expeced from a bin ha is wice as wide as an ideal bin (i.e., 1 LSB differenial nonlineariy). As he bins ge narrower wih a higher precision converer, he densiy can be linearized, bu ys is an approximaion. The saisically correc mehod o measure he nonlineariies is o esimae he ransiions from he daa. Then he differenial nonlineariy is he difference beween adjacen ransiion levels minus 1 LSB. The inegral nonlineariy is he difference beween he esimaed ransiion level and he ideal ransiion level. n (2.6) and (2.3), P(V~, V~) is replaced by he measured frequency of occurrence H/N using he frequency subsiuion principle and hen solved for V~, wich is. an esimae of V6. n solving (2.6), he offse VOcan be eliminaed since i only shifs fib and V.. does no affec he inegral or differenial ~nonlineariy. Thus, he simpler (2.3) can be solved for Vb. Taking he cosine of boh sides of (2.3) and using he following ideniies yields (4.10): fi -(2@s(%i~b cos(a /3) =cos(a)cos(/3)+ sin(a) sin(fi) (1 cos sin l = A ((4.8) (4.9) -A2(1-c0s2(H+v 410) The quadraic equaion (4.10) can be solved for fib. n he soluion, he posiive square roo erm is used so ha lfi~ is greaer han Va: fib=vacos(y)+sin( y)lq. (!,J1) This gives ~~ in erms of V.. n general, fi=~_lcos(*)+sin(*)i~ (4.12)
5 824 EEE JOURNAL OF SOLD-STATE CRCUTS, VOL. SC-19, NO 6, DECEMBER 1984 Raher han a recursive formulaion ha is subjec o cumulaive errors, ~ can be compued direcly by using he boundary condiion V. = A and using a cumulaive ljsogram C17(i ) of i bins insead of he ih hisogram bin H(i): () ~= Aces mch( ~ i ) (4.13) A is no known, bu being a linear facor, all ransiions ~ can be normalized o A so ha he full range of ransiions is +1. To esimae he inegral nonlineariy wih he same precision as he differenial nonlineariy, many more samples and a much longer esing ime are required. Thus, drifs in he ADC volage reference and he sine wave oscillaor s ampliude and offse volage can give erroneous resuls. The FFT es is no sensiive o hese problems since very few samples are needed. V. FFT NTEGRAL NONLNEARTY TEST The discree Fourier ransform compued wih a fas Fourier ransform algorihm can be used o measure he nonlineariy of he ADC ransfer funcion. The seup is as before, bu his ime he daa aken are no pu in a hisogram. They are jus sored in he sequence aken, sen o a UNX file, and hen Fourier ransformed. The specrum of he oupu will conain he inpu sine wave, quanizaion error, and any harmonic disorion caused by inegral nonlineariy. The heoreical signal-onoise raio is (6r + 1.8) db [4]. f he harmonic disorion is more han 6n db below he fundamenal ampliude, he error caused by inegral nonlineariy can be concluded o be less han 1 bi and herefore negligible. The inpu frequency mus be chosen so ha harmonics aliased ino he baseband do no add o he fundamenal. The raw daa from he ADC were modified by a Harming window [5] o reduce he effecs of runcaing a sine wave before a FFT. f he sampled daa conairi an inegral number of periods of he inpu sine wave, he FFT will be accurae. f he samples conain a fracion of a sine wave period, he FFT will have gross disorions. V. TESTNG FOR SPECFC APPLCATONS The specific applicaion and nonlineariy errors of he ADC should dicae he ype of es o be performed. f he applicaion is for insrumenaion, he quaniy o be esed is differenial and inegral nonlineariy so he code densiy es is appropriae. f he use is in a digial audio sysem, he appropriae ess would be in he frequency domain. The FFT would be inerpreed for harmonic disorion, frequency response, S/N, ec. The code densiy es is mos sensiive o differenial nonlineariy errors, while an FFT es is mos sensiive o inegral nonlineariy errors. Thus, he ype of error o be 0.1, 1.- ; (.!.: -0.3 / z [ ~.- & al Oupu (a) 1 1. A - -. g..- : 0.50 d p f \ ~ / G Oupu Fig bi, CMOS, remsor-sring ADC, (a) Differenial nonlineany. (b) negral nonlineariy. measured, raher han he applicaion of he be a facor in deermining which es o use. V. (b) CODE DENSTY TEST RESULTS \ ADC, \ would Three differen designs of A/D converers were esed. All were of he successive approximaion variey. The firs was an 8 bi, resisor-sring, CMOS converer, he second a 12 bi, bipolar, laser-rimmed, R-2R ladder converer, and he hird a 15 bi, CMOS, self-calibraing ADC wih a capacior firay and resisor sring. For he 8 bi, resisor-sring ADC, samples corresponding o a 0.1 bi precision wih 99 percen confidence were aken. The differenial nonlineariies and inegral nonlineariies are shown in Fig. 4(a) and (b). There are no differenial nonlineariies greaer han 1/4 bi; hus, he inegral nonlineariy is smooh and is never greaer han 2 bis. Manufacurers will ofen pass a bes-fi-line hrough his inegral nonlineariy plo and claim + 1 LSB inegral nonlineariy wih a gain and offse error. There is no paern o he errors ha are from random mismaches in he resisor sring. Wih only 5000 samples, he inegral nonlineariy is no longer smooh, bu has he same shape and approximaely he same wors case error. However, he differenial nonlineariy has a large degree of uncerainy, bu he major nonlineariies are visible. For he 12 bi, R-2R ADC, he major carries are clearly visible where he inegral nonlineariy jumps 1 bi. The differenial nonlineariy in Fig. 5(a) shows large spikes ha
6 DOERNBERG e al.: FULL-SPEED TESTNG OF A/D CONVERTERS m w ~ G : 6-1. oo~ ~~, oo~ Oupu (a) ,1000 Frequency (Hz) (a),:r -60 ii, Oupu (b) Fig bi, bipolar, laser-rimmed, R-2 R ADC. Noe he spikes in he differenial nonlineariy and jumps in he inegraf nonlineariy a he major carry ransiions. (a) Differenial nonlineariy. (b) negral nonlineany. -80 ~! _ mn!- o Frequency (b) Fig. 7. FFT specrum for a 12 bi ADC sampling a 495 Hz sine wave. (a) 4096-poin FFT. (b) 1024-poin FFT. ( Hz) g 10 : !-J F\g. 8. Classical mehod of ADC esing. The inegraor is forced (.o he ransiion volage and measured by he DVM Oupu Fig. 6. Differenial nonlineariy for 15 bi, CMOS, self-calibraing ADC wih capacior array main DAC and resisor-sring sub-dac. correspond o resisor mismaches. The oher errors appear periodic since he resisors wih unrimmed, random errors are used repeaedly over he range of he ADC. This is in conras o he resisor sring where each resisor is used once; hence, he errors are no periodic. The 15 bi self-calibraing ADC wih capacior-array main DAC and resisor-sring sub-dac differenial nonlineariy plo is shown in Fig. 6. A. FFT Tes Resuls The 12 bi R-2R ADC was used for he FFT es. Fig. 7(a) is a 4096-poin FFT of a 495 Hz sine wave sampled a 8012 Hz. The harmonics are clearly visible 72 db below he fundamenal, corresponding o 12 bi inegral lineariy. This is wihin he 1 bi inegral nonlineariy specified for he converer. n Fig. 7(b), 1024 samples are used. Since each sample in he ime domain corresponds o one poin in he frequency domain, he feaures are less clear. The decreasing number of samples again increases he noise level, as would a less precise A/D converer. V. COMPARSON TO CLASSCAL TESTNG A classical ADC es is shown concepually in Fig. 8 [6]. The inegraor is driven o each ransiion and held a ha volage while a compuer-conrolled DVM measures he ransiion poin. This is an exremely slow process since he inegraor loop mus sele and hen he DVM akes a reading. The firs drawback o his esis ha he accuracy of he es depends on he DVM. More imporan is ha his is a saic es of he ADC. The AQC is measuring a dc volage, no a high-frequency inpu. Mos converers are esed his way, bu hey claim he same characerisics and accuracy for a maximum conversion rae dynamic inpu. There is no measuremen of dynamic errors. Wih he hisogram and FFT ess, he inpu can be as high a frequency as desired o es for frequency-dependen errors.
7 826 EEE JOURNAL OF SOLD-STATE CRCUTS, VOL. SC-19, NO. 6, DECEMBER 1984 p ogp 00 1F!!% % m ~Q 01 m 05 R=.05 A -05 Q025 z a= oo a=, _- a, ~o a i -1 5 QO , log(n) MNMUM NUMBER OF SAMPLES PER BN, N Fig. 9. Minimum number of samples needed per bin for differenial nonlineariy es. The parameers are ~, he precision of he esimae, and a, he confidence level. Tesing a high-precision converer by he classical mehod can be in error due o noise a he ADC inpu. Bu he hisogram es being saisical and sampling each bin many imes raher han once will average ou any random noise. The precision of he classical esis limied by he DVM. Bu in he hisogram es, aking more samples increases he precision. Exending he classical es o higher precision converers is again limied by DVM precision and accuracy. Wih a hisogram es, he inpu source mus be known o more precision han he ADC and can be easily verified wih a specrum analyzer. Lasly, he minicompuer mus have enough memory o sore 2 hisogram bins. The inegraor loop akes approximaely 5 s o measure each ransiion or 5 2/3 h o compleely es a 12 bi ADC. f a precision DAC is used insead of an inegraor, he speed should increase by a facor of 10 o abou 30 in, which is sill very slow. Wih a hisogram of 1000 couns per bin, for 99 percen confidence wih 0.1 bi precision, i will ake 9 rnin o ake he daa a a 8 khz inpu rae. For producion esing, he confidence level and precision can be reduced o 95 percen and 0.25 bi precision, decreasing he number of samples needed and he esing ime by a facor of 10. Fig. 9 shows he radeoffs among confidence level, precision, and he number of samples required. The esing ime can also be reduced by aking he daa faser since he rae is currenly limied by he minicompuer, no he ADC under es. X. D/A CONVERTER TESTNG To es D/A converers, a dual of he hisogram es is sough. This would be a number generaor inpu o he DAC and a device quanizing he analog oupu and couning he number of occurrences of each oupu o ge a hisogram. Bu he quanizaion is done by an ADC, and has he same disadvanages as using a DAC o es he ADC S, ha is, speed, precision, and noise. However, a dual of he FFT es is an analog specrum analysis. npu a digial since wave o he DAC and look a he specrum. deally, here will be he fundamenal, quanizaion noise, and harmonic disorion. The level of harmonic disorion is relaed o he nonlineariy of he DAC ransfer curve jus as inegral nonlineariy in he ADC was deduced from a FFT. X. SUMMARY The code densiy es produces a hisogram of he digial oupu codes of an ADC sampling a known inpu. The code densiy is used o compue he volage ransiion levels ha characerize he ADC. This es is compleely general in ha i ess high-precision and high-speed converers. is superior o a radiional ransiion es since i is done a full speed wih a dynamic inpu and he resuls do no depend on he accuracy of a DAC or DVM. FFT ess are performed o measure he inegral nonlineariy, disorion, and signal-o-noise raio. Unlike classical es mehods, he mehods proposed here also es he sampleand-hold and can measure he inernal noise of he ADC. D/A converers can be esed by a dual of he FFT es, using a digial sine wave inpu and an analog specrum analyzer. APPENDX The uncerainy in he differenial nonlineariy is he uncerainy in he widh of he bin ~+ ~ ~. From (4.13), = A (( COS 7r(CH(i)+H(i)) N, _C05 % Czl(i) ( N, )} ) (Al) (A2) CH(i ) is he oal number of codes in bins 1 hrough i and ACH(i) = H(i + 1) is he number of codes in bin [i + 1]. Now define F(X) = cos TX/N: q+l ~= A[F(CH(i)+ ACH(i))-F(CH(i))] = -A[F(CH(i)+ ACH(i))-F(CH(i))] ~_ ACH( i ) (A3).ACH( i ) (A4) AdF(CH(Z))AcH(i) dch( i ) () _ AmACH(i). rch(i) sm N, N (A5) (A6) ACH(i) and CH(i) are random variables, bu we can assume ha CH( i ) is known since i only affecs he inegral nonlineariy, and his confidence inerval is for he differenail nonlineariy. Thus, he random variable is
8 DOERNBERG e d.: FULL-SPEED TESTNG OF A/D CONVERTERS 827 ACH(Z) and ~+ ~ ~ is of he form spending o a zero inpu or bin [2 1] is used. 7i-CH( i ) ~ p(2n-1) =!_ Y= A~sin ~. (A7) J sin-l [~]-sin-l[-~]) (i 2 v ref Le he random variable ACH(i) = X and be disribued sin l (A14) v wih mean px and sandard deviaion u, wih he following [1A2* noaion: X - (px, OX). f Y= ax+ b, hen Y - (up. + For V,ef = A, his reduces o (2/n), sin 1 [1/2 ]. For any b, aux). reasonable value of n, he sin 1 argumen is small, and sin 1(x) = x so p = P(2-1) =1/7i-2 -l. The condiion is ~=a~x+b=(a:sin(w )))px A ) ha z:, TCH(i) N,> uy=aux=az sin (A9) N, () N, 0 b (A15) Now he mean px and sandard deviaion OXof he random variable ACH( i ) are needed o find p ~ and UY. Any given sample wih eiher go in bin [i] or i will no go in bin [i]. This is a wo oucome, or Bernoulli rial, wi{h [11 binomial disribuion characerized by mean np(l p) = np [21 since p << 1 and sandard The oal number of samples aken is n = N, and p is he probabil- [31 iy ha a sample goes in a bin. Thus, ACH( i ) - B( rp, G). From (A8) and (A9), py = pam sin [1N, TCH( i ) TCH( i ) fial?ln - [ y 1 [4] [51 (A1O) 6] (All) REFEMNCES Dynamic performance esing of A o D converer Hewle Packard Produc Noe 5180A-2. R. A. Belcher, Audio non-lineariy: An iniial appraisaf of a double comb-filer mehod of measuremen, BBC Res. Dep., Rep., 1$~77/40, Nov H. Murvei, An inegraed-circui based speech recogniion sysem, Ph.D. disseraion, Univ. of California, Berkeley, Dec W. A. Keser, Characerizing and esing A/D and D/A converers for color video applicaions; EEE Trans. Circuis Sys., vol. CAS-25, pp , July F. J. Harris, <On he use of windows for hwmonic analysis wih he discree Fourier ransform; Proc. EEE, vol. 66, Jan J. J. Corcoran, T. Homak, and P. B. Skov, A high-resoluion error ploer for analog-o-digird converer% EEE Tram.,wrurn. A4eus., vol. M-24, Dec f he number of samples is large, he binomial disribuion can be approximaed by a Normal or Gaussian disribuion and can be found for any choice of a. Z.l is he number of sandard deviaions which can be found from a abulaed lising of he sandard normal disribuion for any chosen alpha. Thus, he measured bi widh p y, which is nominally 1 bi, lies wihin is rue value wih olerance Z@y wih 100(1 a) percen confidence. Thus, ~@Y < ~PY. PPY is he olerance o which he bi widh is known. Subsiuing (A1O) and (All) for o, and p,, Z:,2 NJ32 p (A13) Joey Doemberg (S 84) was born in Los Angeles, CA, on Ocober 28, He received he B.S. (highes honors) and M.S. degrees in elcricaf engineering from he Universiy of Cd[fomia, Berkeley; in 1981 and 1983, respecively. He is currenly a Research Assisan a UC Berkeley, working for he Ph.D. degree in he area of inegraed circui, high-speed A/D converers for video applicaions, He is Asc doing research on compuer-aided A/D and D/ A converer esing. Mr. Doernberg is a member of Phi Bea Kappa, Ea Kappa Nu, and Tau Bea Pi. David A. Hodges (S 59-M65-SM71- F 77), for a phoograph and biography, see his issue, p p is he probabiliy of a sample going in a bin and is a Hae%eung Lee (S 84), for a phoograph and biography, see his issue, p. funcion of he bin [i], so he minimum p, P(2n -1) corre- 819.
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